# Hexagonal tiling honeycomb

Hexagonal tiling honeycomb :
| Perspective projection view within Poincaré disk model ||
| Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
| Schläfli symbols | {6,3,3} t{3,6,3} 2t{6,3,6} 2t{6,3$^{[3]}$} t{3$^{[3,3]}$} |
| Coxeter diagrams | ↔ ↔ ↔ ↔ |
| Cells | {6,3} |
| Faces | hexagon {6} |
| Edge figure | triangle {3} |
| Vertex figure | tetrahedron {3,3} |
| Dual | Order-6 tetrahedral honeycomb |
| Coxeter groups | ${\overline {V}}_{3}$, [3,3,6] ${\overline {Y}}_{3}$, [3,6,3] ${\overline {Z}}_{3}$, [6,3,6] ${\overline {VP}}_{3}$, [6,3$^{[3]}$] ${\overline {PP}}_{3}$, [3$^{[3,3]}$] |
| Properties | Regular |
In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.
The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex.

## Images

Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {∞,3} of H$^2$, with horocycles circumscribing vertices of apeirogonal faces.
| {6,3,3} | {∞,3} |
| One hexagonal tiling cell of the hexagonal tiling honeycomb | An order-3 apeirogonal tiling with a green apeirogon and its horocycle |

## Symmetry constructions

It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: [6,3,3], [3,6,3], [6,3,6], [6,3$^{[3]}$] and [3$^{[3,3]}$] , having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)$^*$] (remove 3 mirrors, index 24 subgroup); [3,6,3$^*$] or [3$^*$,6,3] (remove 2 mirrors, index 6 subgroup); [1$^+$,6,3,6,1$^+$] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3$^{[3,3]}$]. The ringed Coxeter diagrams are , , , and , representing different types (colors) of hexagonal tilings in the Wythoff construction.

## Related polytopes and honeycombs

The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
11 paracompact regular honeycombs :
| {6,3,3} | {6,3,4} | {6,3,5} | {6,3,6} | {4,4,3} | {4,4,4} |
| {3,3,6} | {4,3,6} | {5,3,6} | {3,6,3} | {3,4,4} |
It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb.
[6,3,3] family honeycombs :
| {6,3,3} | r{6,3,3} | t{6,3,3} | rr{6,3,3} | t$_{0,3}${6,3,3} | tr{6,3,3} | t$_{0,1,3}${6,3,3} | t$_{0,1,2,3}${6,3,3} |
| {3,3,6} | r{3,3,6} | t{3,3,6} | rr{3,3,6} | 2t{3,3,6} | tr{3,3,6} | t$_{0,1,3}${3,3,6} | t$_{0,1,2,3}${3,3,6} |
It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures.
{p,3,3} honeycombs :
| Space | Space | S$^3$ | S$^3$ | S$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ |
| Form | Form | Finite | Finite | Finite | Paracompact | Noncompact | Noncompact | Noncompact |
| Name | Name | {3,3,3} | {4,3,3} | {5,3,3} | {6,3,3} | {7,3,3} | {8,3,3} | ... {∞,3,3} |
| Image | Image | | | | | | | |
| Coxeter diagrams | 1 | | | | | | | |
| Coxeter diagrams | 4 | | | | | | | |
| Coxeter diagrams | 6 | | | | | | | |
| Coxeter diagrams | 12 | | | | | | | |
| Coxeter diagrams | 24 | | | | | | | |
| Cells {p,3} | Cells {p,3} | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} |
It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of hexagonal tiling cells:
{6,3,p} honeycombs vte :
| Space | H$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ |
| Form | Paracompact | Paracompact | Paracompact | Paracompact | Noncompact | Noncompact | Noncompact |
| Name | {6,3,3} | {6,3,4} | {6,3,5} | {6,3,6} | {6,3,7} | {6,3,8} | ... {6,3,∞} |
| Coxeter | | | | | | | |
| Image | | | | | | | |
| Vertex figure {3,p} | {3,3} | {3,4} | {3,5} | {3,6} | {3,7} | {3,8} | {3,∞} |

### Rectified hexagonal tiling honeycomb

Rectified hexagonal tiling honeycomb :
| Type | Paracompact uniform honeycomb |
| Schläfli symbols | r{6,3,3} or t$_1${6,3,3} |
| Coxeter diagrams | ↔ |
| Cells | {3,3} r{6,3} or |
| Faces | triangle {3} hexagon {6} |
| Vertex figure | triangular prism |
| Coxeter groups | ${\overline {V}}_{3}$, [3,3,6] ${\overline {P}}_{3}$, [3,3$^{[3]}$] |
| Properties | Vertex-transitive, edge-transitive |
The rectified hexagonal tiling honeycomb, t$_1${6,3,3}, has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The half-symmetry construction alternates two types of tetrahedra.
| Hexagonal tiling honeycomb | Rectified hexagonal tiling honeycomb or |
| Related H$^2$ tilings ||
| Order-3 apeirogonal tiling | Triapeirogonal tiling or |

### Truncated hexagonal tiling honeycomb

Truncated hexagonal tiling honeycomb :
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | t{6,3,3} or t$_{0,1}${6,3,3} |
| Coxeter diagram | |
| Cells | {3,3} t{6,3} |
| Faces | triangle {3} dodecagon {12} |
| Vertex figure | triangular pyramid |
| Coxeter groups | ${\overline {V}}_{3}$, [3,3,6] |
| Properties | Vertex-transitive |
The truncated hexagonal tiling honeycomb, t$_{0,1}${6,3,3}, has tetrahedral and truncated hexagonal tiling facets, with a triangular pyramid vertex figure.
It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t{∞,3} with apeirogonal and triangle faces:

### Bitruncated hexagonal tiling honeycomb

Bitruncated hexagonal tiling honeycomb Bitruncated order-6 tetrahedral honeycomb :
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | 2t{6,3,3} or t$_{1,2}${6,3,3} |
| Coxeter diagram | ↔ |
| Cells | t{3,3} t{3,6} |
| Faces | triangle {3} hexagon {6} |
| Vertex figure | digonal disphenoid |
| Coxeter groups | ${\overline {V}}_{3}$, [3,3,6] ${\overline {P}}_{3}$, [3,3$^{[3]}$] |
| Properties | Vertex-transitive |
The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t$_{1,2}${6,3,3}, has truncated tetrahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.

### Cantellated hexagonal tiling honeycomb

Cantellated hexagonal tiling honeycomb :
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | rr{6,3,3} or t$_{0,2}${6,3,3} |
| Coxeter diagram | |
| Cells | r{3,3} rr{6,3}×{3} |
| Faces | triangle {3} square {4} hexagon {6} |
| Vertex figure | wedge |
| Coxeter groups | ${\overline {V}}_{3}$, [3,3,6] |
| Properties | Vertex-transitive |
The cantellated hexagonal tiling honeycomb, t$_{0,2}${6,3,3}, has octahedron, rhombitrihexagonal tiling, and triangular prism cells, with a wedge vertex figure.

### Cantitruncated hexagonal tiling honeycomb

Cantitruncated hexagonal tiling honeycomb :
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | tr{6,3,3} or t$_{0,1,2}${6,3,3} |
| Coxeter diagram | |
| Cells | t{3,3} tr{6,3}×{3} |
| Faces | triangle {3} square {4} hexagon {6} dodecagon {12} |
| Vertex figure | mirrored sphenoid |
| Coxeter groups | ${\overline {V}}_{3}$, [3,3,6] |
| Properties | Vertex-transitive |
The cantitruncated hexagonal tiling honeycomb, t$_{0,1,2}${6,3,3}, has truncated tetrahedron, truncated trihexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.

### Runcinated hexagonal tiling honeycomb

Runcinated hexagonal tiling honeycomb :
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | t$_{0,3}${6,3,3} |
| Coxeter diagram | |
| Cells | {3,3} {6,3}×{6}×{3} |
| Faces | triangle {3} square {4} hexagon {6} |
| Vertex figure | irregular triangular antiprism |
| Coxeter groups | ${\overline {V}}_{3}$, [3,3,6] |
| Properties | Vertex-transitive |
The runcinated hexagonal tiling honeycomb, t$_{0,3}${6,3,3}, has tetrahedron, hexagonal tiling, hexagonal prism, and triangular prism cells, with an irregular triangular antiprism vertex figure.

### Runcitruncated hexagonal tiling honeycomb

Runcitruncated hexagonal tiling honeycomb :
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | t$_{0,1,3}${6,3,3} |
| Coxeter diagram | |
| Cells | rr{3,3}x{3}x{12} t{6,3} |
| Faces | triangle {3} square {4} dodecagon {12} |
| Vertex figure | isosceles-trapezoidal pyramid |
| Coxeter groups | ${\overline {V}}_{3}$, [3,3,6] |
| Properties | Vertex-transitive |
The runcitruncated hexagonal tiling honeycomb, t$_{0,1,3}${6,3,3}, has cuboctahedron, triangular prism, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

### Runcicantellated hexagonal tiling honeycomb

Runcicantellated hexagonal tiling honeycomb runcitruncated order-6 tetrahedral honeycomb :
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | t$_{0,2,3}${6,3,3} |
| Coxeter diagram | |
| Cells | t{3,3}x{6} rr{6,3} |
| Faces | triangle {3} square {4} hexagon {6} |
| Vertex figure | isosceles-trapezoidal pyramid |
| Coxeter groups | ${\overline {V}}_{3}$, [3,3,6] |
| Properties | Vertex-transitive |
The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t$_{0,2,3}${6,3,3}, has truncated tetrahedron, hexagonal prism, and rhombitrihexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

### Omnitruncated hexagonal tiling honeycomb

Omnitruncated hexagonal tiling honeycomb Omnitruncated order-6 tetrahedral honeycomb :
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | t$_{0,1,2,3}${6,3,3} |
| Coxeter diagram | |
| Cells | tr{3,3}x{6}x{12} tr{6,3} |
| Faces | square {4} hexagon {6} dodecagon {12} |
| Vertex figure | irregular tetrahedron |
| Coxeter groups | ${\overline {V}}_{3}$, [3,3,6] |
| Properties | Vertex-transitive |
The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t$_{0,1,2,3}${6,3,3}, has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with an irregular tetrahedron vertex figure.